Answer 1

it is a lightweight open source database system used for Internet sites by the most, since a couple of years it's owned by Oracle and its future is uncertain but its fathers forked the project creating MariaDB which could replace mysql in the future in case Oracle "kills" mysql.
despite this situation Mysql is still the best solution (give a chance to MariaDB) if you want to develop a "simple" Internet site (afak Google still relies a lot on it) but if you prefer security, for instance for a shop or payment system, I would suggest you Oracle (xe version is free from charge) or, with more effort for maintenance, postresql

Answer 2

simply, a program for using/running SQL language.